On geometry of numbers and uniform rational approximation to the Veronese curve

Abstract

We refine upper bounds for the classical exponents of uniform approximation for a linear form on the Veronese curve in dimension from 3 to 9. For dimension three, this in particular shows that a bound previously obtained by two different methods is not sharp. Our proof involves parametric geometry of numbers and investigation of geometric properties of best approximation polynomials. Slightly stronger bounds have been obtained by Poels with a different method contemporarily. In fact, we obtain the same bounds as a conditional result.

Upper bounds for the uniform simultaneous Diophantine exponents

“…In 1969, Davenport and Schmidt [6] verified that λ𝑛 (𝜉) ⩽ 1∕⌊ 𝑛 2 ⌋ which, in view of the lower bound 1∕𝑛 ⩽ λ𝑛 (𝜉), leaves a small window for the values of the exponent λ𝑛 . Later, their bound was slightly improved by several authors, most recent results belong to Laurent [7] and Schleischitz [13]. However, all these upper bounds are still of the form…”

“…In 1969, Davenport and Schmidt [6] verified that $$\widehat{\lambda }_n(\xi ) \leqslant 1/\lfloor \frac{n}{2}\rfloor$$ which, in view of the lower bound $$1/n\leqslant \widehat{\lambda }_n(\xi )$$, leaves a small window for the values of the exponent $$\widehat{\lambda }_n$$. Later, their bound was slightly improved by several authors, most recent results belong to Laurent [7] and Schleischitz [13]. However, all these upper bounds are still of the form $$\frac{2}{n} + O(n^{-2})$$.…”